Rational models of the complement of a subpolyhedron in a manifold with boundary

TL;DR

This paper develops algebraic models to understand the rational homotopy type of complements of subpolyhedra in manifolds with boundary, leading to invariance results for configuration spaces and explicit models for certain cases.

Contribution

It constructs algebraic models for the rational homotopy type of complements in manifolds with boundary, extending previous work to new settings and providing explicit models.

Findings

Rational homotopy invariance of configuration spaces under certain conditions

Explicit models for configuration spaces in specific manifolds

Construction of algebraic models from pair maps under high codimension

Abstract

Let W be a compact simply connected triangulated manifold with boundary and $K \subset W$ be a subpolyhedron. We construct an algebraic model of the rational homotopy type of the complement $W \setminus K$ out of a model of the map of pairs $(K, K \cap \partial W) \to (W,\partial W)$ under some high codimension hypothesis. We deduce the rational homotopy invariance of the configuration space of two points in a compact manifold with boundary under 2-connectedness hypotheses. Also, we exhibit nice explicits models of these configuration spaces for a large class of compact manifolds.